Integrand size = 34, antiderivative size = 155 \[ \int \frac {\cos ^2(c+d x) (A+A \sec (c+d x))}{\sqrt {a-a \sec (c+d x)}} \, dx=\frac {11 A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{4 \sqrt {a} d}-\frac {2 \sqrt {2} A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a-a \sec (c+d x)}}\right )}{\sqrt {a} d}+\frac {5 A \sin (c+d x)}{4 d \sqrt {a-a \sec (c+d x)}}+\frac {A \cos (c+d x) \sin (c+d x)}{2 d \sqrt {a-a \sec (c+d x)}} \] Output:
11/4*A*arctan(a^(1/2)*tan(d*x+c)/(a-a*sec(d*x+c))^(1/2))/a^(1/2)/d-2*2^(1/ 2)*A*arctan(1/2*a^(1/2)*tan(d*x+c)*2^(1/2)/(a-a*sec(d*x+c))^(1/2))/a^(1/2) /d+5/4*A*sin(d*x+c)/d/(a-a*sec(d*x+c))^(1/2)+1/2*A*cos(d*x+c)*sin(d*x+c)/d /(a-a*sec(d*x+c))^(1/2)
Time = 0.33 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.77 \[ \int \frac {\cos ^2(c+d x) (A+A \sec (c+d x))}{\sqrt {a-a \sec (c+d x)}} \, dx=\frac {A \left (\sqrt {1+\sec (c+d x)} (5 \sin (c+d x)+\sin (2 (c+d x)))+11 \text {arctanh}\left (\sqrt {1+\sec (c+d x)}\right ) \tan (c+d x)-8 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\sec (c+d x)}}{\sqrt {2}}\right ) \tan (c+d x)\right )}{4 d \sqrt {1+\sec (c+d x)} \sqrt {a-a \sec (c+d x)}} \] Input:
Integrate[(Cos[c + d*x]^2*(A + A*Sec[c + d*x]))/Sqrt[a - a*Sec[c + d*x]],x ]
Output:
(A*(Sqrt[1 + Sec[c + d*x]]*(5*Sin[c + d*x] + Sin[2*(c + d*x)]) + 11*ArcTan h[Sqrt[1 + Sec[c + d*x]]]*Tan[c + d*x] - 8*Sqrt[2]*ArcTanh[Sqrt[1 + Sec[c + d*x]]/Sqrt[2]]*Tan[c + d*x]))/(4*d*Sqrt[1 + Sec[c + d*x]]*Sqrt[a - a*Sec [c + d*x]])
Time = 0.95 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.08, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.382, Rules used = {3042, 4510, 27, 3042, 4510, 27, 3042, 4408, 3042, 4261, 216, 4282, 216}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\cos ^2(c+d x) (A \sec (c+d x)+A)}{\sqrt {a-a \sec (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A \csc \left (c+d x+\frac {\pi }{2}\right )+A}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \sqrt {a-a \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 4510 |
| \(\displaystyle \frac {A \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a-a \sec (c+d x)}}-\frac {\int -\frac {\cos (c+d x) (5 a A+3 a \sec (c+d x) A)}{2 \sqrt {a-a \sec (c+d x)}}dx}{2 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\cos (c+d x) (5 a A+3 a \sec (c+d x) A)}{\sqrt {a-a \sec (c+d x)}}dx}{4 a}+\frac {A \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a-a \sec (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {5 a A+3 a \csc \left (c+d x+\frac {\pi }{2}\right ) A}{\csc \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a-a \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{4 a}+\frac {A \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a-a \sec (c+d x)}}\) |
| \(\Big \downarrow \) 4510 |
| \(\displaystyle \frac {\frac {5 a A \sin (c+d x)}{d \sqrt {a-a \sec (c+d x)}}-\frac {\int -\frac {11 A a^2+5 A \sec (c+d x) a^2}{2 \sqrt {a-a \sec (c+d x)}}dx}{a}}{4 a}+\frac {A \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a-a \sec (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {11 A a^2+5 A \sec (c+d x) a^2}{\sqrt {a-a \sec (c+d x)}}dx}{2 a}+\frac {5 a A \sin (c+d x)}{d \sqrt {a-a \sec (c+d x)}}}{4 a}+\frac {A \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a-a \sec (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {11 A a^2+5 A \csc \left (c+d x+\frac {\pi }{2}\right ) a^2}{\sqrt {a-a \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}+\frac {5 a A \sin (c+d x)}{d \sqrt {a-a \sec (c+d x)}}}{4 a}+\frac {A \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a-a \sec (c+d x)}}\) |
| \(\Big \downarrow \) 4408 |
| \(\displaystyle \frac {\frac {16 a^2 A \int \frac {\sec (c+d x)}{\sqrt {a-a \sec (c+d x)}}dx+11 a A \int \sqrt {a-a \sec (c+d x)}dx}{2 a}+\frac {5 a A \sin (c+d x)}{d \sqrt {a-a \sec (c+d x)}}}{4 a}+\frac {A \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a-a \sec (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {16 a^2 A \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a-a \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+11 a A \int \sqrt {a-a \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{2 a}+\frac {5 a A \sin (c+d x)}{d \sqrt {a-a \sec (c+d x)}}}{4 a}+\frac {A \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a-a \sec (c+d x)}}\) |
| \(\Big \downarrow \) 4261 |
| \(\displaystyle \frac {\frac {16 a^2 A \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a-a \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {22 a^2 A \int \frac {1}{\frac {a^2 \tan ^2(c+d x)}{a-a \sec (c+d x)}+a}d\frac {a \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}}{d}}{2 a}+\frac {5 a A \sin (c+d x)}{d \sqrt {a-a \sec (c+d x)}}}{4 a}+\frac {A \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a-a \sec (c+d x)}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {16 a^2 A \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a-a \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {22 a^{3/2} A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{d}}{2 a}+\frac {5 a A \sin (c+d x)}{d \sqrt {a-a \sec (c+d x)}}}{4 a}+\frac {A \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a-a \sec (c+d x)}}\) |
| \(\Big \downarrow \) 4282 |
| \(\displaystyle \frac {\frac {\frac {22 a^{3/2} A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{d}-\frac {32 a^2 A \int \frac {1}{\frac {a^2 \tan ^2(c+d x)}{a-a \sec (c+d x)}+2 a}d\frac {a \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}}{d}}{2 a}+\frac {5 a A \sin (c+d x)}{d \sqrt {a-a \sec (c+d x)}}}{4 a}+\frac {A \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a-a \sec (c+d x)}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {\frac {22 a^{3/2} A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{d}-\frac {16 \sqrt {2} a^{3/2} A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a-a \sec (c+d x)}}\right )}{d}}{2 a}+\frac {5 a A \sin (c+d x)}{d \sqrt {a-a \sec (c+d x)}}}{4 a}+\frac {A \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a-a \sec (c+d x)}}\) |
Input:
Int[(Cos[c + d*x]^2*(A + A*Sec[c + d*x]))/Sqrt[a - a*Sec[c + d*x]],x]
Output:
(A*Cos[c + d*x]*Sin[c + d*x])/(2*d*Sqrt[a - a*Sec[c + d*x]]) + (((22*a^(3/ 2)*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/d - (16*Sqrt [2]*a^(3/2)*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a - a*Sec[c + d* x]])])/d)/(2*a) + (5*a*A*Sin[c + d*x])/(d*Sqrt[a - a*Sec[c + d*x]]))/(4*a)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(b/d) Subst[Int[1/(a + x^2), x], x, b*(Cot[c + d*x]/Sqrt[a + b*Csc[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S ymbol] :> Simp[-2/f Subst[Int[1/(2*a + x^2), x], x, b*(Cot[e + f*x]/Sqrt[ a + b*Csc[e + f*x]])], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_ .) + (a_)], x_Symbol] :> Simp[c/a Int[Sqrt[a + b*Csc[e + f*x]], x], x] - Simp[(b*c - a*d)/a Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] /; F reeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(f*n)), x] - Simp[1/(b*d *n) Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)*Simp[a*A*m - b*B* n - A*b*(m + n + 1)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, m}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && LtQ[n, 0]
Time = 5.32 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.48
| method | result | size |
| default | \(\frac {A \sqrt {2}\, \sin \left (d x +c \right ) \left (32 \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right )+\left (6 \cos \left (d x +c \right )^{3}+27 \cos \left (d x +c \right )^{2}+4 \cos \left (d x +c \right )+15\right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+\left (33 \cos \left (d x +c \right )+33\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}{2}\right )+\left (48 \cos \left (d x +c \right )+48\right ) \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right )\right )}{24 d \sqrt {-a \left (-1+\sec \left (d x +c \right )\right )}\, \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \left (\cos \left (d x +c \right )^{2}+2 \cos \left (d x +c \right )+1\right )}\) | \(229\) |
Input:
int(cos(d*x+c)^2*(A+A*sec(d*x+c))/(a-a*sec(d*x+c))^(1/2),x,method=_RETURNV ERBOSE)
Output:
1/24*A/d*2^(1/2)*sin(d*x+c)*(32*2^(1/2)*(-cos(d*x+c)/(1+cos(d*x+c)))^(1/2) *cos(d*x+c)+(6*cos(d*x+c)^3+27*cos(d*x+c)^2+4*cos(d*x+c)+15)*(-2*cos(d*x+c )/(1+cos(d*x+c)))^(1/2)+(33*cos(d*x+c)+33)*2^(1/2)*arctan(1/2*2^(1/2)*(-2* cos(d*x+c)/(1+cos(d*x+c)))^(1/2))+(48*cos(d*x+c)+48)*arctan(1/2*2^(1/2)/(- cos(d*x+c)/(1+cos(d*x+c)))^(1/2)))/(-a*(-1+sec(d*x+c)))^(1/2)/(-cos(d*x+c) /(1+cos(d*x+c)))^(1/2)/(cos(d*x+c)^2+2*cos(d*x+c)+1)
Time = 0.10 (sec) , antiderivative size = 462, normalized size of antiderivative = 2.98 \[ \int \frac {\cos ^2(c+d x) (A+A \sec (c+d x))}{\sqrt {a-a \sec (c+d x)}} \, dx=\left [\frac {8 \, \sqrt {2} A a \sqrt {-\frac {1}{a}} \log \left (-\frac {2 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}} \sqrt {-\frac {1}{a}} - {\left (3 \, \cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right )}{{\left (\cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) - 11 \, A \sqrt {-a} \log \left (\frac {2 \, {\left (\cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}} - {\left (2 \, a \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right )}{\sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) - 2 \, {\left (2 \, A \cos \left (d x + c\right )^{3} + 7 \, A \cos \left (d x + c\right )^{2} + 5 \, A \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}}}{8 \, a d \sin \left (d x + c\right )}, \frac {8 \, \sqrt {2} A \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) - 11 \, A \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) - {\left (2 \, A \cos \left (d x + c\right )^{3} + 7 \, A \cos \left (d x + c\right )^{2} + 5 \, A \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}}}{4 \, a d \sin \left (d x + c\right )}\right ] \] Input:
integrate(cos(d*x+c)^2*(A+A*sec(d*x+c))/(a-a*sec(d*x+c))^(1/2),x, algorith m="fricas")
Output:
[1/8*(8*sqrt(2)*A*a*sqrt(-1/a)*log(-(2*sqrt(2)*(cos(d*x + c)^2 + cos(d*x + c))*sqrt((a*cos(d*x + c) - a)/cos(d*x + c))*sqrt(-1/a) - (3*cos(d*x + c) + 1)*sin(d*x + c))/((cos(d*x + c) - 1)*sin(d*x + c)))*sin(d*x + c) - 11*A* sqrt(-a)*log((2*(cos(d*x + c)^2 + cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c) - a)/cos(d*x + c)) - (2*a*cos(d*x + c) + a)*sin(d*x + c))/sin(d*x + c) )*sin(d*x + c) - 2*(2*A*cos(d*x + c)^3 + 7*A*cos(d*x + c)^2 + 5*A*cos(d*x + c))*sqrt((a*cos(d*x + c) - a)/cos(d*x + c)))/(a*d*sin(d*x + c)), 1/4*(8* sqrt(2)*A*sqrt(a)*arctan(sqrt(2)*sqrt((a*cos(d*x + c) - a)/cos(d*x + c))*c os(d*x + c)/(sqrt(a)*sin(d*x + c)))*sin(d*x + c) - 11*A*sqrt(a)*arctan(sqr t((a*cos(d*x + c) - a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c)))* sin(d*x + c) - (2*A*cos(d*x + c)^3 + 7*A*cos(d*x + c)^2 + 5*A*cos(d*x + c) )*sqrt((a*cos(d*x + c) - a)/cos(d*x + c)))/(a*d*sin(d*x + c))]
\[ \int \frac {\cos ^2(c+d x) (A+A \sec (c+d x))}{\sqrt {a-a \sec (c+d x)}} \, dx=A \left (\int \frac {\cos ^{2}{\left (c + d x \right )}}{\sqrt {- a \sec {\left (c + d x \right )} + a}}\, dx + \int \frac {\cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{\sqrt {- a \sec {\left (c + d x \right )} + a}}\, dx\right ) \] Input:
integrate(cos(d*x+c)**2*(A+A*sec(d*x+c))/(a-a*sec(d*x+c))**(1/2),x)
Output:
A*(Integral(cos(c + d*x)**2/sqrt(-a*sec(c + d*x) + a), x) + Integral(cos(c + d*x)**2*sec(c + d*x)/sqrt(-a*sec(c + d*x) + a), x))
\[ \int \frac {\cos ^2(c+d x) (A+A \sec (c+d x))}{\sqrt {a-a \sec (c+d x)}} \, dx=\int { \frac {{\left (A \sec \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{2}}{\sqrt {-a \sec \left (d x + c\right ) + a}} \,d x } \] Input:
integrate(cos(d*x+c)^2*(A+A*sec(d*x+c))/(a-a*sec(d*x+c))^(1/2),x, algorith m="maxima")
Output:
integrate((A*sec(d*x + c) + A)*cos(d*x + c)^2/sqrt(-a*sec(d*x + c) + a), x )
Time = 0.47 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.91 \[ \int \frac {\cos ^2(c+d x) (A+A \sec (c+d x))}{\sqrt {a-a \sec (c+d x)}} \, dx=\frac {\frac {8 \, \sqrt {2} A \arctan \left (\frac {\sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a}}{\sqrt {a}}\right )}{\sqrt {a}} - \frac {11 \, A \arctan \left (\frac {\sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a}}{2 \, \sqrt {a}}\right )}{\sqrt {a}} - \frac {\sqrt {2} {\left (3 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{\frac {3}{2}} A + 10 \, \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a} A a\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{2}}}{4 \, d} \] Input:
integrate(cos(d*x+c)^2*(A+A*sec(d*x+c))/(a-a*sec(d*x+c))^(1/2),x, algorith m="giac")
Output:
1/4*(8*sqrt(2)*A*arctan(sqrt(a*tan(1/2*d*x + 1/2*c)^2 - a)/sqrt(a))/sqrt(a ) - 11*A*arctan(1/2*sqrt(2)*sqrt(a*tan(1/2*d*x + 1/2*c)^2 - a)/sqrt(a))/sq rt(a) - sqrt(2)*(3*(a*tan(1/2*d*x + 1/2*c)^2 - a)^(3/2)*A + 10*sqrt(a*tan( 1/2*d*x + 1/2*c)^2 - a)*A*a)/(a*tan(1/2*d*x + 1/2*c)^2 + a)^2)/d
Timed out. \[ \int \frac {\cos ^2(c+d x) (A+A \sec (c+d x))}{\sqrt {a-a \sec (c+d x)}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^2\,\left (A+\frac {A}{\cos \left (c+d\,x\right )}\right )}{\sqrt {a-\frac {a}{\cos \left (c+d\,x\right )}}} \,d x \] Input:
int((cos(c + d*x)^2*(A + A/cos(c + d*x)))/(a - a/cos(c + d*x))^(1/2),x)
Output:
int((cos(c + d*x)^2*(A + A/cos(c + d*x)))/(a - a/cos(c + d*x))^(1/2), x)
\[ \int \frac {\cos ^2(c+d x) (A+A \sec (c+d x))}{\sqrt {a-a \sec (c+d x)}} \, dx=-\sqrt {a}\, \left (\int \frac {\sqrt {-\sec \left (d x +c \right )+1}\, \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )}{\sec \left (d x +c \right )-1}d x +\int \frac {\sqrt {-\sec \left (d x +c \right )+1}\, \cos \left (d x +c \right )^{2}}{\sec \left (d x +c \right )-1}d x \right ) \] Input:
int(cos(d*x+c)^2*(A+A*sec(d*x+c))/(a-a*sec(d*x+c))^(1/2),x)
Output:
- sqrt(a)*(int((sqrt( - sec(c + d*x) + 1)*cos(c + d*x)**2*sec(c + d*x))/( sec(c + d*x) - 1),x) + int((sqrt( - sec(c + d*x) + 1)*cos(c + d*x)**2)/(se c(c + d*x) - 1),x))